Fusion Operators in the Generalized $τ^{(2)}$-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin

0.32 MB·English

by  Shi-shyr Roan

7
0
0
2

n
a
J

1 Fusion Operators in the Generalized τ(2)-model and
3
Root-of-unity Symmetry of the XXZ Spin Chain of

]
h Higher Spin
1
c
e
m
-
t Shi-shyr Roan
a
t Institute of Mathematics
s
.
t Academia Sinica
a
m Taipei , Taiwan
- (email: [email protected] )
d
n
o
c
[

3
v
8
Abstract
5
2
We constructthe fusionoperatorsinthegeneralizedτ(2)-modelusingthefusedL-operators,
7
0 and verify the fusion relations with the truncation identity. The algebraic Bethe ansatz discus-
6
sion is conducted on two special classes of τ(2) which include the superintegrable chiral Potts
0
/ model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and
t
a demonstrate that the sl -loop-algebra symmetry exists for the root-of-unity XXZ spin chain
m 2
with a higher spin, where the evaluation parameters for the symmetry algebra are identified
-
d by the explicit Fabricius-McCoy current for the Bethe states. Parallels are also drawn to the
n
comparison with the superintegrable chiral Potts model.
o
c
:
v
i 1999 PACS: 05.50.+q, 02.20.Tw, 75.10Jm
X
2000 MSC: 17B65, 39B72, 82B23
r
a Key words: τ(2)-model, Chiral Potts model, XXZ spin chain, Bethe ansatz, Fusion operator
1In this paper, the term ”XXZ spin chain” means a lattice model with the L-operator associated to the trigono-
metricR-matrixoftheice-typemodel. The”XXZspin chainofhigherspin”hereisnamedasthe”XXZHeisenberg
model with arbitrary spin” in [31].
1
1 Introduction
In this paper, we investigate the uniform structure about symmetry of various lattice models2,
namely the generalized τ(2)-model, the chiral Potts model (CPM), and XXZ spin chain of higher
spin. Parallelsarealsodrawntodifferentiatingthesymmetrynatureappearedinthosemodels. The
generalizedτ(2)-model,alsoknownastheBaxter-Bazhanov-Stroganov (BBS)model[3,7,9,10,24],
isthesix-vertex modelwithaparticularfield(see, [7]page3), i.e. havingtheR-matrixoftheasym-
metricsix-vertexmodel,(see(2.3)inthispaper). Notethatthenon-symmetricdiagonalBoltzmann
weights of this twisted R-matrix differentiates it from the usual (gauge transformed) trigonomet-
ric R-matrix (see, for instance [11] (3.3) and references therein), and the usual construction of
L-operator by employing the highest-weight representation theory of quantum groups or algebras
fails in this context. However using the cyclic N-vector representation of the finite Weyl algebra,
one can construct a five-parameter family of L-operators for the N-state generalized τ(2)-model.
When the parameters are restricted on a family of high genus curves, called the rapidities, one
obtains the τ(2)-matrix of the solvable N-state CPM. In [9] Bazhanov and Stroganov showed that
thecolumn-transfer-matrixoftheL-operatorinCPMpossessespropertiesoftheBaxter’s Q-matrix
(i.e., thechiral-Potts-transfer matrix)withsymmetryoperatorssimilarto, butnotexactlythesame
as, the sl -loop algebra. Indeed the τ(2)-degenerate eigenvalue spectrum occurs only in superin-
2
tegrable case [39], where the Onsager-algebra-symmetry operators in the quantum Hamiltonian
chain [25] derived from the Baxter’s Q-matrix provide the precise symmetry structure of the su-
perintegrable τ(2)-model [40]. Due to the lack of difference property of rapidities, the study of
CPM, such as the calculation of eigenvalues [4, 5, 34] and order parameter [8], relies on the method
of functional relations among the τ(j)- and chiral-Potts-transfer matrices, consisting of (recursive
and truncation) T-fusion relation, TQ- and QQ-relations. On the other hand, in the study of
the ”zero-field” six-vertex model ([6] Sect. 3) with the usual trigonometric R-matrix, it has been
extensively analyzed in [15, 18, 19, 41] that the degeneracy of the spin-1 XXZ Hamiltonian occurs
2
whentheanisotropic parameter q is aroot of unity with theextra sl -loop-algebra symmetry of the
2
system [15]. Recently, the sl -loop-algebra symmetry appears again in the XXZ chain of spin-N 1
2 2−
at the Nth root of unity [36]3. Results about the root-of-unity symmetry of the six-vertex model
were first discovered by algebraic Bethe ansatz and quantum group theory, later by the method
of Baxter’s Q-operator, and the degeneracy of eigenvalues was further found in the root-of-unity
eight-vertex model[16,17,20]. Bythiseffort, FabriciusandMcCoy observedthesimilarity between
the root-of-unity eight-vertex model and CPM, and proposed the conjectural functional relations
of the eight-vertex model by encoding the root-of-unity symmetry property in a proper Q-operator
of the theory, as an analogy to functional relations in CPM [21]. Along this line, the Q-operator
incorporated with the sl -loop-algebra symmetry of the spin-1 XXZ chain at the roots of unity was
2 2
constructed in [42] wherefunctional relations in the Fabricius-McCoy comparison were verified. By
2All themodels discussed in this paper will always assume with theperiodic condition.
3TheXXZchainofspin-N2−1 attheNthrootofunityinthispaperisphrasedastheNilpotentBazhanov-Stroganov
model in [36].
2
this,weintendtomakeadetailed investigation aboutcommonfeatures showninthoseknownmod-
els, compare the symmetry structure, then further explore more unknown models in the scheme of
functional relations. Among the function relations, the fusion relation with ”truncation” property
plays a vital role in connecting the T-and Baxter’s Q-matrix. Usually onecan derive theboundary
fusion relation, i.e. the ”truncation” identity, using the Baxter’s TQ-relation for any Q-operator
(not necessarily satisfying the QQ-relation) if a Q-matrix exists (see, e.g. [2, 29, 37]). Conversely,
the validity of boundary fusion relation may as well strongly suggests the existence of Baxter’s
Q-operator in many cases (see, e.g. [35] and references therein). The models we shall discuss in
this article are the generic generalized τ(2)-model (where no Q-operator is known), superintegrable
CPM, and XXZ spin chain of higher spin at the roots of unity. In the present paper, we obtain
two main results related to these models. First, by constructing the fusion matrix from the explicit
fused L-operator, we derive the fusion relations with the truncation identity for these models with-
out relying on the theory of Baxter’s Q-operator. Second, by the algebraic Bethe ansatz of XXZ
spin chain of higher spin at the roots of unity and certain special τ(2)-models which include the
superintegrable CPM, we show the sl -loop-algebra symmetry of the root-of-unity XXZ chain for
2
a higher spin as in the spin-1 case [15, 18, 19]. Furthermore, they all share the same structure as
2
the superintegrable CPM about the Bethe equation and the evaluation (Drinfeld) polynomial for
the symmetry algebra as in [41, 42]. As a consequence, a conjecture raised in [36] about the simple
polynomial property for the sl -loop-algebra evaluation parameters in the XXZ chain of spin-N 1
2 2−
has been justified. Note that for the τ(2)-model in the generic case where algebraic Bethe ansatz
cannot beapplied dueto the lack of peudovacuum state, our fusion-matrix studystrongly suggests,
with computational evidences in cases, that thefusion relations with thetruncation identity always
hold. Hence by [24], the separation-of-variables method provides the solution of Baxter equation
associated to the τ(2)-model.
The quantum inverse scattering method/algebraic Bethe ansatz developed by the Leningrad
school in the early eighties [23, 28, 33] systematized earlier results about the Bethe ansatz of
two-dimensional lattice models in an algebraic scheme by using the Yang-Baxter (YB) equation
as a central role of solvability. A YB solution defines a local L-operator, which gives rise to the
algebra of quantum monodromy matrices, called the ABCD-algebra. From the ABCD-algebra, one
can construct a set of commuting transfer matrices, which in principal could be simultaneously
diagonalized using a basis derived from the pseudovacuum state by the Bethe-ansatz technique.
Furthermore, one can define the quantum determinant of the algebra, a concept first introduced in
[26] (or see [28] Chapter VIII), and played an important role in deriving the fused transfer matrices
in this work. It is known that this algebraic method has long been used in the investigation of
XXZ spin chains (see e.g., [31, 44] references therein), and in the τ(2)-model [45, 46]. For the XXZ
spin chain of higher spin in the root-of-unity case, it possesses some extra symmetry carrying the
”evaluation” parameters, which indeed determine the eigenvalues of fusion matrices and implicitly
encode symmetry of the model. Hence a new structure, not seen in the general XXZ chain of
higher spin, appears in the root-of-unity theory. In this article, we employ the ABCD-algebra
method in the generalized τ(2)-model and the root-of-unity XXZ chain of higher spin to study the
3
fusion matrix through some explicit fused L-operators. The boundary fusion relation will be our
main concern. The technique is first to make use of the quantum determinant of L-operators, not
only on the explicit form, but also its nature in commuting the fusion-product of elements so that
the recursive fusion relation holds. Next the detailed analysis about ”averaging” the L-operator
leads to the boundary fusion relation. Since the ABCD-algebra of generalized τ(2)-model carries
a non-equivalent, though similar, structure as the algebra for the XXZ spin chain due to the non-
symmetric Boltzmann weights in the R-matrix of τ(2)-model, we shall provide a more elaborate
discussionaboutthealgebraicBetheansatzoftheτ(2)-model(thoughmanylikeroutineexercises in
the field). This is because the correct formulation with the precise expression of physical quantities
is non-trivial, and required for the later CPM algebraic-Bethe-ansatz discussion in this work when
comparingitwiththecompleteresultsofsuperintegrableCPMderivedfromthefunctionalrelations
[1, 4, 5, 10]. It is also needed for the parallel symmetry discussion between superintegrable N-state
CPM and the XXZ chain of spin-N 1. The algebraic Bethe ansatz is known to be applied to the
2−
superintegrable τ(2)-model; however to what extent the results obtained by the algebraic-Bethe-
ansatz method compared with the complete τ(2)-eigenvalues andits degeneracy knownin the study
ofCPM[1,5,40]hasnotbeenfullydiscussedintheliteraturetothebestoftheauthor’sknowledge,
especially about possible symmetry structures of the model. To this end, we propose a scheme for
certain special classes of generalized τ(2)-model, with the superintegrable CPM included, where
the pseudovacuum state exists so that the algebraic-Bethe-ansatz technique can be performed in
the way like the root-of-unity XXZ spin chain. When applying to the superintegrable τ(2)-model,
the setting enables us to conduct exact investigations of various problems. One can rediscover
the Bethe equation, fusion relations, forms of eigenvalue spectrum and evaluation polynomials,
known in the theory of CPM. Furthermore, certain eigenvectors derived from the pseudovacuum
state can also be extracted by the algebraic Bethe ansatz method. Nevertheless, only certain
sectors of the spectrum are covered by this scheme. In the case of the root-of-unity XXZ spin
chain of higher spin, the algebraic Bethe ansatz method produces the correct form of evaluation
polynomial for the degeneracy by a detailed analysis of the eigenvalues of fusion matrices. Then
thezero-averages ofoff-diagonal elements inthequantummonodromymatrix, correspondingtothe
vanishing property of the N-string creation operator, give rise to the sl -loop-algebra symmetry of
2
the root-of-unity six vertex model by a ”q-scaling” procedure in [15]. Thereupon one can identify
the evaluation polynomial of sl -loop-algebra representation for a Bethe state through the explicit
2
Fabricius-McCoy current ([21] (1.37)) of the model.
This paper is organized as follows. In section 2, we discuss the fusion relations of the general-
ized τ(2)-model. We begin with some preparatory work in subsection 2.1 on the algebraic structure
derived from YB relation for the generalized τ(2)-model [45, 46]. Using standard techniques in
the ABCD algebra and quantum determinant for the twisted R-matrix, we construct in subsection
2.2 the fusion operators from the fused L-operators so that the recursive fusion relation holds.
By studying the average of L-operators, we then show evidences, verified in cases by direct com-
putations, that the boundary fusion relation is valid for the generalized τ(2)-model in subsection
2.3. In section 3, we study two special classes of BBS models, which include the superintegrable
4
CPM, by the algebraic-Bethe-ansatz method where the pseudo-vacuum exists. We then perform
the investigation on the Bethe equation and Bethe states for such models in subsection 3.1. The
algebraic-Bethe-ansatz discussion of special BBS models when restricted on the superintegrable
τ(2)-model recovers the Bethe equation and evaluation polynomial of Onsager-algebra symmetry in
the superintegrable CPM [1, 5, 40]. The comparison of those algebraic-Bethe-ansatz results with
the complete results known in the theory of superintegrable CPM is given in subsection 3.2. In
section 4, we study the root-of-unity symmetry of XXZ spin chain with a higher spin. First we
briefly review some basic concepts in the algebraic Bethe ansatz of XXZ spin chain that are needed
for later discussions, (for more detailed information, see e.g., [23, 33] references therein). Then we
summarize results in [27, 30, 32, 35, 42, 43] about the fusion relation for the spin-1 XXZ chain at
2
roots of unity. Using the fused L-operators, we extend the construction of fusion operators in the
spin-1 to the spin-d 1 XXZ chain at Nth root of unity for 2 d N in subsection 4.1, where
2 −2 ≤ ≤
the fusion relations are derived. Furthermore through the fusion-matrix-eigenvalue discussion, we
extract the correct form of evaluation polynomial incorporated with the Bethe equation. In sub-
section 4.2, we show that the root-of-unity XXZ chain of spin-d 1 possesses the sl -loop-algebra
−2 2
symmetry, and verify the evaluation polynomial by the explicit Fabricius-McCoy current of Bethe
states. In subsection 4.3, we make the comparison between the root-of-unity XXZ chain of spin-
N 1 and the N-state superintegrable CPM, which are known to be closely related in literature
2−
[1, 5, 36]. We close in section 5 with some concluding remarks.
2 Fusion Relations and Algebraic Bethe Ansatz of Generalized
τ(2)-model
We first briefly review some basic structures in the ABCD-algebra for the generalized τ(2)-model
in subsection 2.1. Then in subsection 2.2 we construct the fusion operators as the trace of fused
L-operators so that the recursive fusion relation holds, and the boundary fusion relation will be
discussed in subsection 2.3.
2.1 ABCD-algebra and quantum determinant in the generalized τ(2)-model
We start with some basic notions about algebraic structures in the generalized τ(2)-model. The
summary will be sketchy, but also serve to establish notations, (for more detailed information, see
[45] and references therein).
For a positive integer N, we fix the Nth root of unity, ω = e2π√N−1. Denote by CN the vector
space of N-cyclic vectors with n as the standard basis where Z = Z/NZ , and X,Z the
CN-operators defined byX n {=| in}+n∈1Z,NZ n = ωn n for n Z , whichNsatisfy the Weyl relation,
N
| i | i | i | i ∈
XZ = ω 1ZX, with XN = ZN = 1. The L-operator of the generalized τ(2)-model is built upon
−
5
the Weyl operators X,Z with C2-auxiliary space and CN-quantum space4 :
1+tκX (γ δX)Z A(t) B(t)
L(t) = − =: , t C, (2.1)
t(α−βX)Z−1 tαγ+ βκδX ! C(t) D(t) ! ∈
where α,β,γ,δ,κ C are parameters, which satisfy the YB equation5
∈
R(t/t)(L(t) 1)(1 L(t))= (1 L(t))(L(t) 1)R(t/t ), (2.2)
′ ′ ′ ′
aux aux aux aux
O O O O
for the R-matrix of the asymmetric six-vertex model,
tω 1 0 0 0
−
 0 t 1 ω 1 0 
R(t) = − − . (2.3)
 0 t(ω 1) (t 1)ω 0 
 − − 
 
 0 0 0 tω 1 
 − 
 
Then the monodromy matrix for the quantum chain of size L,
L
A (t) B (t)
L (t) = L (t) L (t) = L L , L (t) := L(t) at site ℓ, (2.4)
ℓ 1 L ℓ
ℓ=1 ⊗···⊗ CL(t) DL(t) !
O
again satisfy the YB equation (2.2), and the ω-twisted trace,
τ(2)(t) := A (ωt)+D (ωt),
L L
L
form a family of commuting operators of the L-tensor space CN of CN. We shall denote the
⊗
L
spin-shift operator of CN again by X(:= L X ) if no confusion could arise, which carries the
⊗ ℓ=1 ℓ
Z -charge, denoted by Q = 0,...,N 1. By
N Q
−
[X,A ]= [X,D ]= 0, XB = ω 1B X, XC = ωC X, (2.5)
L L L − L L L
X commutes with the τ(2)-matrix. The relation (2.2) for the monodromy matrix gives rise to
an algebra structure of the operator-entries A (t),B (t),C (t),D (t), called the ABCD-algebra,
L L L L
(algebra of quantum monodromy matrix), in which the following conditions hold:
[A(t),A(t )] = [B(t),B(t)] = [C(t),C(t)] = [D(t),D(t)] = 0;
′ ′ ′ ′
(tω t)A(t)B(t )= (t t)B(t)A(t)+t(ω 1)A(t )B(t), A,B C,D;
′ ′ ′ ′ ′
− − − −→
(tω t)B(t)A(t )= (t t)ωA(t)B(t)+t(ω 1)B(t)A(t), A,B C,D; (2.6)
′ ′ ′ ′ ′ ′
− − − −→
(tω t)C(t)A(t) = (t t)A(t)C(t)+t(ω 1)C(t)A(t ), A,C B,D;
′ ′ ′ ′ ′ ′
− − − −→
(tω t)A(t)C(t) = (t t)ωC(t)A(t)+t(ω 1)A(t)C(t), A,C B,D.
′ ′ ′ ′ ′
− − − −→
4Here we use the form of L-operator in accord with the convention used in [10, 40], which is essentially the
transpose of theL-operator in [24, 45].
5Notethat(2.1) satisfy theYBrelation (2.2) aswellforageneral ω notnecessary aroot ofunity,usingtheWeyl
operators X,Z with XZ =ω−1ZX.
6
The quantum determinant follows from the ABCD algebra by setting t = ωt:
′
B(ωt)A(t) = A(ωt)B(t), D(ωt)C(t)= C(ωt)D(t);
A(t)C(ωt) = ωC(t)A(ωt), B(t)D(ωt)= ωD(t)B(ωt),
(2.7)
det ( L )(t) := D(ωt)A(t) C(ωt)B(t)= A(ωt)D(t) B(ωt)C(t)
q ℓ
− −
= A(t)D(ωt) ωC(t)B(ωt)= D(t)A(ωt) ω 1B(t)C(ωt).
N −
− −
or equivalently, the quantum determinant of the monodromy matrix (2.4) is characterized by rank-
one property of R(ω 1) in the following relation,
−
R(ω 1)( L (t) 1)(1 L (ωt)) = (1 L (ωt))( L (t) 1)R(ω 1)= det ( L )(t) R(ω 1),
− ℓ ℓ ℓ ℓ − q ℓ −
⊗ ⊗ ⊗ ⊗ ⊗ ·
aux aux aux aux
O O O O
(2.8)
with the explicit form for det ( L )(t):
q ℓ
⊗
βδ
det ( L )(t) = q(t)LXL, q(t) := +(αδ+ωβγ)t+ωαγκt2. (2.9)
q ℓ
⊗ κ
The third- and fifth relations of (2.6) yield
A(t)B(s) = ωt(−tωss)B(s)A(t)+ (ωω(−t 1s))tB(t)A(s), D(t)B(s)= ωω(tt−ss)B(s)D(t)− (ωω(−t 1s))tB(t)D(s).
− − − −
By moving B(t )’s to the left hand side of A(t),D(t), one obtains
i
DA((tt))QQmimi==11BB((ttii)) −+==PQQmkmkmimi====1111ωωωωωt((((−ωω((ttttt−−−−−ω−−tttt11ttiiii))kk))tt))··QQQmimi==mimi==1111,,ii6=6=BBkk((ttωωiitω(())ktttAD−kkk−−−ω((ttttttiiii)))) ··BB((tt))Qmimi==11,,ii6=6=kkBB((ttii))AD((ttkk)),. (2.10)
P Q Q
Similarly the second- and fourth relations in (2.6) yield
AD((tt))QQmimi==11CC((ttii)) +−==PQQmkmkmimi====1111ωt((tt−ωωt−−tt−ω−−−−ttttii11ttiikk))tt··kkQQQmimi==mimi==1111CC,,ii6=6=((ttkkii))ωtADttktkk−k(−−(−tωttt))ttiiii ··CC((tt))Qmimi==11,,ii6=6=kkCC((ttii))AD((ttkj)),. (2.11)
Notethatbyscaling thet-varPiable, parameQtersin(2.1) can bereQducedtothecaseα+γ = 0, among
which, with one more constraint ωβ+δ = 0, one can express α= γ = y 1, β = ω 1δ = µ2xy 2,
− − −
− −
κ = µ2y 2 for (x,y,µ) C3. For the N-state CPM, the rapidity variables of L-operator (2.1)
−
− ∈
are defined by
kxN = 1 k µ N, kyN = 1 k µN, (x,y,µ) C3, (2.12)
′ − ′
− − ∈
where k ,k are temperature-like parameters with k2 + k2 = 1. In the superintegrable case, the
′ ′
parameters in (2.1) and the quantum determinant are given by
α= β = γ = ω 1δ = κ= 1, det L(t)= ωh2(t)X, (2.13)
− q
− − −
(see, e.g. [40] Sect. 5). Hereafter we shall always use h(t) to denote
h(t) := 1 t. (2.14)
−
7
2.2 Fused L-operator in generalized τ(2)-model
Hereweconstruct thefusedL-operator L(j)(t)for thefusionτ(j)-matrix with L(2)(t) = L(t)in(2.1).
For convenience of notations, we shall also denote the standard basis 1 of the C2-auxiliary
|± i
space of L(t), and its dual basis by x = 1 , y = 1 ; x = 1 , y = 1. For non-negative
| i |− i h | h− |
integers m,n, we denote by xmyn the completely symmetric (m+n)-tensor of C2 defined by
b b
m+n xmyn = xb b... x y ... y+ all other terms by permutations,
n ! ⊗ ⊗ ⊗ ⊗ ⊗
m n
b b b b b b
similarly for xmyn. For j 1|, the{zCj-}aux|iliary{zspac}e is the space of completely symmetric (j 1)-
≥ −
tensors of the C2, with the following canonical basis e(j) and the dual basis e(j)∗:
k k
j 1 k
e(kj) = xj−1−kyk, e(kj)∗ = −k− !xj−1−kyk, k = 0,...,j −1. (2.15)
b b
TheL(j)(t)istheoperator L(j)(t) withthe Cj-auxiliary and CN-quantumspace, where
k,l
(cid:18) (cid:19)0 k,l j 1
L(j)(t) is expressed by ≤ ≤ −
k,l
L(kj,l)(t) = he(kj)∗|L(ωj−2t)⊗aux ···⊗L(ωt)⊗auxL(t)|e(lj)i. (2.16)
ThenL(j)(t)areintertwinedbysomeR(j)-matrix. WithL(j)(t)asthelocaloperator,itsmonodromy
L
matrix defines the commuting family of τ(j)-operators of CN,
⊗
L
τ(j)(t) = tr ( L(j)(ωt)). (2.17)
Cj ℓ
ℓ=1
O
We now show the fusion relation between τ(j+1), τ(j) and τ(j 1) through the quantum determinant
−
(2.9).
j+1
Consider the auxiliary-space tensor C2 Cj as a subspace of C2 with the identification
⊗ ⊗
1 j 1 j 1
(j+1) (j) (j)
e = − x e + − y e , k = 1,...,j 1,
k+1 k+j1 (cid:18) k+1! ⊗ k+1 k ! ⊗ k (cid:19) − −
(cid:0) (cid:1) b b
(j 1) (j) (j) (j+1) (j 1)
and denote f − := x e y e for 0 k j 2. Then e ,f − form a basis of
k ⊗ k+1 − ⊗ k ≤ ≤ − l k
C2 Cj with the dual basis e(j+1)∗,f(j−1)∗ expressed by
⊗ l k
b b
e(kj++11)∗ = x⊗e(kj+)∗1+y⊗e(kj)∗, fk(j−1)∗ = ( 1j ) j−k1 x⊗e(kj+)∗1− kj−+11 y⊗e(kj)∗ .
k+1 (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
(j+1) (j+1)
Then the expression of e ,e ∗ yields
k l
he(kj+1)∗|L(j+1)(t)|e(lj+1)i = he(kj+1)∗|L(ωj−1t)⊗auxL(j)(t)|e(lj+1)i. (2.18)
In order to determine the rest entries of L(ωj 1t) L(j)(t), we need the following simple lemma.
− aux
⊗
8
Lemma 2.1 The second equality in (2.8) is equivalent to the following relations:
x2 L(ωt) L(t)x y = y2 L(ωt) L(t)x y = 0,
aux aux
h | ⊗ | ∧ i h | ⊗ | ∧ i (2.19)
x y L(ωt) L(t)x y = y xL(ωt) L(t) x y = 1det L(t),
h ⊗ | ⊗aux | ∧ i h ⊗ | ⊗aux |− ∧ i 2 q
b b b b
where x y = 1(x y y xb). Hbence we have b b
∧ 2 ⊗ − ⊗
b bhe(k3)∗b|L(ωbt)⊗bauxLb(t)|x⊗yi = he(k3)∗|L(ωt)⊗auxL(t)|y⊗xi for k =0,1,2.
As a consequence, for an integer jb 2b, and v = x or y for 1 ib jb 1, we have
i
≥ ≤ ≤ −
he(kj)∗|L(ωj−2t)⊗aux···⊗auxL(ωtb)⊗aubxL(t)|v1⊗v2⊗···⊗vj−1i (2.20)
= he(kj)∗|L(ωj−2t)⊗aux···⊗auxL(ωt)⊗auxL(t)|vσ1 ⊗vσ2 ⊗···⊗vσj 1i
−
for 0 k j 1, and all permutations σ.
≤ ≤ −
2
Since the entries of L(ωj 1t) L(j)(t) are determined by those of L(ωj 1t) L(t), by
− aux − aux aux
⊗ ⊗ ···⊗
(2.20) one has
e(j+1) L(ωj 1t) L(j)(t)f(j) = 0, 1 k j 1, 0 k j 2. (2.21)
h k+1 | − ⊗aux | k′ i − ≤ ≤ − ≤ ′ ≤ −
Using (2.19) and (2.20), one finds
hfk(j−1)∗|L(ωj−1t)⊗auxL(j)(t)|fk(j′−1)i = hfk(j−1)∗|L(ωj−1t)⊗auxL(j)(t)|(2x∧y)⊗xj−k′−2⊗yk′i
= ( 1j )h( j−k1 x⊗y− kj−+11 y⊗x)⊗e(kj−1)∗|L(ωj−1t)⊗auxL(j)(t)|(2x∧y)⊗xj−k′−2⊗yk′i
k+1 b b b b
=detqL(ω(cid:0)j−2(cid:1)t)·he(kj−1(cid:0))∗|L((cid:1)j−1)(t)|xj−k′−2⊗yk′i, b b b b
which, by (2.9) and (2.19), in turn ybields b
hfk(j−1)∗|L(ωj−1t)⊗auxL(j)(t)|fk(j′−1)i= he(kj−1)∗|L(j−1)(t)|e(kj′−1)iq(ωj−2t)X. (2.22)
From the definition (2.17) of τ(j)-matrices, the relations, (2.18) (2.22) and (2.21), imply the follow-
ing result:
Proposition 2.1 The τ(j)-matrices satisfy the (recursive)fusionrelation bysetting τ(0) = 0,τ(1) =
I,
τ(2)(ωj 1t)τ(j)(t) = z(ωj 1t)Xτ(j 1)(t)+τ(j+1)(t), j 1. (2.23)
− − −
≥
where z(t) = q(t)L with q(t) in (2.9).
2
9
2.3 Boundary fusion relation in generalized τ(2)-model
For convenience, we introduce the following convention for a family of commuting operators O(t):
n 1
[O] (t) := − O(ωit), n Z ,
n 0
∈ ≥
i=0
Y
and the average of O(t) is defined by
O (= O (tN)) = [O] (t).
N
h i h i
The ”classical” L-operator of BBS model is the average of (2.1):
A B 1+( 1)N+1κNtN γN δN
(tN) = h i h i = − − . (2.24)
L C D ! ( 1)N+1(αN βN)tN βNδN +( 1)N+1αNγNtN !
h i h i − − κN −
Theaverages A , B , C , D of the monodromy matrix (2.4) coincide with the classical Lth
L L L L
h i h i h i h i
monodromy associated to (2.24) [45]6:
A B
h Li h Li (tN) = (tN) (tN) (tN)(= (tN)L), (2.25)
1 2 L
CL DL ! L L ···L L
h i h i
By (2.20), the (k,l)th entry L(j)(t) of L(j)(t) is equal to the expression in (2.20) with v = x for
k,l i
1 i j 1 l, and y otherwise. Hence one can express L(j)(t) in terms of entries in (2.1). For
≤ ≤ − −
example, the matrix-form of L(3)(= L(3)(t)) is b
b
[A] (t), A(ωt)B(t), [B] (t)
2 2
 A(ωt)C(t)+C(ωt)A(t), A(ωt)D(t)+C(ωt)B(t), D(ωt)B(t)+B(ωt)D(t) . (2.26)
 [C]2(t), C(ωt)D(t), [D]2(t) 
 
 
Note that one can also write L(3) = A(ωt)B(t),L(3) = D(ωt)A(t)+B(ωt)C(t), L(3) = D(ωt)C(t).
0,1 1,1 1,0
AmongtheL(j)-entriesforageneralj,thefollowingonescanbederivedbysetting v = xj 1 l yl
i − −
⊗ ⊗
in (2.20):
b b
L(j) = [A] (t), L(j) = [B] (t), L(j) = [C] (t), L(j) =[D] (t);
L(0j,0) = [A]j−1 (ωlt)0[,Bj−]1(t), L(jj−)1 = [C]j−1,0 (ωlt)[jD−1] (t), 1j−1l,j−1j 2.j−1 (2.27)
0,l j−l−1 l j−1,l j−l−1 l ≤ ≤ −
By the relation, e(j+1)∗ = e(j)∗ y+e(j)∗ x, for the dual basis of Cj’s, one can compute L(j+1)
k k 1 ⊗ k ⊗ k,l
in terms of L(j)-entries using−the following recursive relations:
L(j+1)(t) = L(j) (ωt)C(t)+L(j)(ωt)A(t),
k,0 k 1,0 k,0 (2.28)
L(j+1)(t) = L(j−) (ωt)D(t)+L(j) (ωt)B(t) for l 1,
k,l k 1,l 1 k,l 1 ≥
− − −
6The formula (2.25) about averages of the monodromy matrix is stated in [45] page 966 as a consequence of
Proposition 5 (ii) there, (orLemma 1.5 in [46]), butwithout proof, and also not with therequired form which could
be misprint in both papers. The correct version should be < △(T) >=< T >< T >, instead of △(< T >) =<
1 2
T >< T >. As the author could not find a proof in literature about the correct Tarasov’s statement, here in
1 2
this paper we provide a mathematical justification about the correct statement in Proposition 2.2 and 4.1 for the
generalized τ(2)-modeland theXXZ spin chain respectively.
10